In a medium saucepan, combine 2 cups cream, milk, 1/2 cup sugar, and instant espresso powder. Cook over medium-low heat for 20 minutes, whisking occasionally.

Fill a large bowl with ice and water to make an ice-water bath; set aside.

In another large bowl, beat egg yolks and remaining 1/2 cup sugar with a whisk for approximately 2 minutes, or until mixture is pale yellow. Pour 1 cup of hot mixture into egg yolks and whisk to combine well. Pour egg mixture back into remaining hot mixture and whisk to combine.

Return saucepan to medium heat. Cook and stir constantly with a wooden spoon for 3 to 5 minutes, or until mixture thickens and holds a line when a finger is drawn across the back of the spoon.

Remove pan from heat and immediately add 1 cup heavy cream and vanilla, stirring to combine. Strain mixture into a large stainless-steel bowl and set bowl in ice-water bath.

Add espresso and remaining 1/2 cup heavy cream, stirring occasionally, until cold. Gently stir in doughnut pieces.

Cover bowl and freeze for 3 to 4 hours, or until very firm, stirring every 30 minutes.